The Fourier version

Because the time axis is well sampled and unaliased, we can safely Fourier transform the data between time t and frequency f:  

$$d (s,r,t) \equiv \int \exp( i 2 \pi f t ) \tilde d(s,r,f) df .$$ (3)

Tildes will indicate Fourier transforms. Transform the slant stack from $\tau_s$ to its frequency f_s:  

$$\tilde S (s,p_s,f_s) = \int \exp( -i 2 \pi f_s \tau_s ) S (s,p_s,\tau_s) d \tau_s .$$ (4)

The slant stack simplifies numerically. Substitute the transform (3) into the slant stack (2), then take the transform (4) of both sides of the equation:  

$$\tilde S (s,p_s ,f_s ) = \int \int \int \exp ( -i 2 \pi f_... ...\pi f (\tau_s + p_s h) ) \tilde d (s,r=s+h,f) dh df d \tau_s.$$ (5)

Rearranging terms, we reduce integrals

$$\begin{eqnarray} \tilde S (s,p_s ,f_s ) &=& \int \int \{ \int \exp[ -i 2 \pi (f... ... &=& \int \exp( i 2 \pi f_s p_s h ) \tilde d (s,r=s+h,f=f_s) dh .\end{eqnarray}$$ (6)

The second step uses the Fourier transform of a delta function $\delta(f) = \int \exp ( -i 2 \pi f t ) dt$.The third uses the behavior of a delta function in an integral $\int g(f) \delta (f-f_0 ) df = g(f_0)$.